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7.2 definite integral — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths7.2 definite integral1 Mark
MCQ
Evaluate $\int\limits_\alpha ^\beta  {\sqrt {\frac{{x - \alpha }}{{\beta  - x}}} } dx$
  • A
    $\frac{\pi }{2}\left( {\alpha  - \beta } \right)$
  • B
    $\frac{\pi }{2}\left( {\alpha  + \beta } \right)$
  • ✓
    $\frac{\pi }{2}\left( {\beta  - \alpha } \right)$
  • D
    $\frac{\pi }{2}\left( {\beta  + \alpha } \right)$
✓

Answer

Correct option: C.
$\frac{\pi }{2}\left( {\beta  - \alpha } \right)$
c
Let $\mathrm{I}=\int_{\alpha}^{\beta} \sqrt{\frac{\mathrm{x}-\alpha}{\beta-\mathrm{x}}} \mathrm{dx}$

$x=\alpha \cos ^{2} t+\beta \sin ^{2} t$

$\mathrm{x}-\alpha=(\beta-\alpha) \sin ^{2} t$

$\beta-\mathrm{x}=(\beta-\alpha) \cos ^{2} \mathrm{t}$

$(\beta-\alpha) \int_{0}^{\pi / 2}(1-\cos 2 t) d t=(\beta-\alpha)\left[t-\frac{\sin 2 t}{2}\right]_{0}^{\pi / 2}=\frac{\pi}{2}(\beta-\alpha)$

$I=\frac{\pi}{2}(\beta-\alpha)$

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